SL2

In differential geometry, the integration along fibers of a k-form yields a $(k-m)$-form where m is the dimension of the fiber, via "integration". More precisely, let $\pi :E\to B$ be a fiber bundle over a manifold with compact oriented fibers. If $\alpha$ is a k-form on E, then let:

$({\pi }_{\ast }\alpha {)}_b(w_1,\dots ,w_{k-m})={\int }_{{\pi }^{-1}(b)}\beta$

where $\beta$ is the induced top-form on the fiber ${\pi }^{-1}(b)$; i.e., an $m$-form given by

$\beta (v_1,\dots ,v_m)=\alpha (\widetilde{w_1},\dots ,\widetilde{w_{k-m}},v_1,\dots ,v_m),\widetilde{w_i}\text{ the lifts of }{w}_i.$

(To see $b\mapsto ({\pi }_{\ast }\alpha {)}_b$ is smooth, work it out in coordinates; cf. an example below.)

${\pi }_{\ast }$ is then a linear map ${\Omega }^k(E)\to {\Omega }^{k-m}(B)$, which is in fact surjective. By Stokes' formula, if the fibers have no boundaries, the map descends to de Rham cohomology:

${\pi }_{\ast }:{\mathrm{H}}^k(E)\to {\mathrm{H}}^{k-m}(B).$

This is also called the fiber integration. Now, suppose $\pi$ is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence $0\to K\to {\Omega }^{\ast }(E)\stackrel{{\pi }_{\ast }}{\to }{\Omega }^{\ast }(B)\to 0$, K the kernel, which leads to a long exact sequence, using ${\mathrm{H}}^k(B)\simeq {\mathrm{H}}^{k+m}(K)$:

$\dots \to {\mathrm{H}}^k(B)\stackrel{\delta }{\to }{\mathrm{H}}^{k+m+1}(B)\stackrel{{\pi }^{\ast }}{\to }{\mathrm{H}}^{k+m+1}(E)\stackrel{{\pi }_{\ast }}{\to }{\mathrm{H}}^{k+1}(B)\to \dots$,

called the Gysin sequence.

Example

Let $\pi :M\times [0,1]\to M$ be an obvious projection. For simplicity, assume $M={ℝ}^n$ with coordinates $x_j$ and consider a k-form:

$\alpha =fd{x}_{i_1}\wedge \dots \wedge d{x}_{i_k}+gdt\wedge d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}}.$

Then, at each point in M,

${\pi }_{\ast }(\alpha )={\pi }_{\ast }(gdt\wedge d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}})=({\int }_0^{1}g(\cdot ,t)dt)d{x}_{j_1}\wedge \dots \wedge d{x}_{j_{k-1}}.$

From this the next formula follows easily: if $\alpha$ is any k-form on $M\times I,$

${\pi }_{\ast }(d\alpha )={\alpha }_1-{\alpha }_0-d{\pi }_{\ast }(\alpha )$

where ${\alpha }_i$ is the restriction of $\alpha$ to $M\times \{i\}$. This formula is a special case of Stokes' formula. As an application of this, let $f:M\times [0,1]\to N$ be a smooth map (thought of as a homotopy). Then the composition $h={\pi }_{\ast }\circ f^{\ast }$ is a homotopy operator:

$d\circ h+h\circ d=f_1^{\ast }-f_0^{\ast }:{\Omega }^k(N)\to {\Omega }^k(M),$

which implies $f_1,f_0$ induces the same map on cohomology. For example, let U be an open ball with center at the origin and let $f_t:U\to U,x\mapsto tx$. Then ${\mathrm{H}}^k(U)={\mathrm{H}}^k(pt)$, the fact known as the Poincaré lemma.

See also

• Gysin homomorphism
• Transgression map

References

• Michele Audin, Torus actions on symplectic manifolds, Birkhauser, 2004